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arXiv · 1004.3956

Regularity of the extremal solution for some elliptic problems with advection

Xue Luo·Dong Ye·Feng Zhou
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Abstract

In this note, we investigate the regularity of extremal solution $u^*$ for semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=λf(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$ is a positive nondecreasing convex function, exploding at a finite value $a\in (0, \infty)$. We show that the extremal solution is regular in low dimensional case. In particular, we prove that for the radial case, all extremal solution is regular in dimension two.

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Xue Luo, Dong Ye, Feng Zhou. 2011-07-06. Regularity of the extremal solution for some elliptic problems with advection. https://arxiv.org/abs/1004.3956

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